Deterministic Distributed Edge-Coloring with Fewer Colors

We present a deterministic distributed algorithm, in the LOCALmodel, that computes a (1+o(1))∆edge-coloring in polylogarithmic-time, so long as the maximum degree ∆ = Ω̃(logn). For smaller ∆, we give a polylogarithmic-time 3∆/2-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of 2∆− 1 colors, and they improve significantly on the recent polylogarithmictime (2∆−1)(1+o(1))-edge-coloring of Ghaffari and Su [SODA’17] and the (2∆−1)-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS’17], positively answering the main open question of the latter. The key technical ingredient of our algorithm is a simple and novel gradual packing of judiciously chosen near-maximum matchings, each of which becomes one of the color classes. Supported by ERC Grant No. 336495 (ACDC).